@Article{AAMM-9-515,
author = {Karlsen , K. H. and Towers , J. D.},
title = {Convergence of Monotone Schemes for Conservation Laws with Zero-Flux Boundary Conditions},
journal = {Advances in Applied Mathematics and Mechanics},
year = {2018},
volume = {9},
number = {3},
pages = {515--542},
abstract = {<p style="text-align: justify;">We consider a scalar conservation law with zero-flux boundary conditions
imposed on the boundary of a rectangular multidimensional domain. We study monotone
schemes applied to this problem. For the Godunov version of the scheme, we
simply set the boundary flux equal to zero. For other monotone schemes, we additionally
apply a simple modification to the numerical flux. We show that the approximate
solutions produced by these schemes converge to the unique entropy solution, in the
sense of [7], of the conservation law. Our convergence result relies on a BV bound on
the approximate numerical solution. In addition, we show that a certain functional
that is closely related to the total variation is nonincreasing from one time level to the
next. We extend our scheme to handle degenerate convection-diffusion equations and
for the one-dimensional case we prove convergence to the unique entropy solution.</p>},
issn = {2075-1354},
doi = {https://doi.org/10.4208/aamm.2016.m-s1},
url = {http://global-sci.org/intro/article_detail/aamm/12162.html}
}